## The monotony and continuity of functions

### A sufficient condition for increasing functions

If at each point of the interval , the function **is increasing** on this interval

### A sufficient condition for decreasing functions

If at each point of the interval , the function **is decreasing** on this interval

**Remark.** These conditions are only sufficient but not necessary conditions for growth and decreasing functions

### A necessary and sufficient condition of constancy of the function

**A function is constant on an interval if and only if when all points of the interval**

## The extrema (maximums and minimums) of the function

### The maximum point

**Definition:** the Point of defining functions is called a maximum point of this function if there is a neighborhood of the point that for all from this neighborhood inequality

— the maximum point

— max

### The point of a minimum

**Definition:** the Point of defining functions is called the minimum point of this function if there is a neighborhood of the point that for all from this neighborhood inequality

— minimum point

— minimum

### The critical point

**Definition:** the Interior points of the domain of definition of functions where the derivative is zero or does not exist are called critical

### Necessary condition for an extremum

— the extremum point or not exists

(but not at each point where or not there will be an extremum!)

### Sufficient condition of extremum

at the point of sign changes at point of maximum

at the point the sign changes at the point of minimum

### An example of a graph of a function that has an extremum

— critical point

## The study of the function on the monotonicity and extrema

- Find the domain of definition and the intervals on which the function is continuous
- To find the derivative
- Find the critical points, i.e. the interior points determining where or not there
- Denote the critical point on the domain of definition, find sign of derivative and nature of function at each interval, which splits the definition area
- For each critical point determine whether it is high or low or is not an extremum point
- Record porni the result of the study (intervals of monotonicity and extrema)

**Example.**

Scope:

The function is continuous at every point of its domain of definition

there is in the entire scope

when

increases with and

subsides when

Extreme points:

Extremes:

## Maximum and minimum values of continuous functions on the interval

**Property:** If the function is continuous on an interval and has therein a finite number of critical points, then it attains its maximum and minimum values on this interval either at a critical point belonging to this interval or at the endpoints of the interval

### Finding maximum and minimum values of continuous functions on the interval

- To find the derivative
- Find the critical points ( or not exists)
- Choose critical points that belong to a given segment
- To calculate the function values at critical points and the endpoints of the interval
- Compare the two values and choose the smallest and largest

**Example.** when

if and when

A given segment belongs to only critical point